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  1. MarkFL

    Practice Evaluate Trigonometric Expressions

    Here is a diagram for \(45^{\circ}\). This triangle is a right isosceles triangle, which means \(x=y\). Use Pythagoras to determine their values...
  2. MarkFL

    Practice Evaluate Trigonometric Expressions

    Consider the following equilateral within the unit circle: Notice that the \(x\)-axis bisects the triangle horizontally. In quadrant I, we have a 30°-60°-90° triangle. As each side of the entire equilateral triangle has a side length of 1 unit, what must the value of \(y\) be? \(y\) is the...
  3. MarkFL

    Practice Evaluate Trigonometric Expressions

    I recommend knowing the following by heart: \sin(0^{\circ})=\sin\left(0\right)=0 \cos(0^{\circ})=\cos\left(0\right)=1 \sin(30^{\circ})=\sin\left(\frac{\pi}{6}\right)=\frac{1}{2} \cos(30^{\circ})=\cos\left(\frac{\pi}{6}\right)=\frac{\sqrt{3}}{2}...
  4. MarkFL

    Practice Three Angles

    I have more time now, so I'll try to outline exactly how I approached this. Consider the following diagram: First I pictured the unit circle, and the line \(y=-\dfrac{1}{2}\). The \(y\)-coordinate of any point on the unit circle represents the sine of the angle required to get to that point...
  5. MarkFL

    Practice Reference Angle

    The reference angle is the smallest angle subtended by a radius \(0\le\theta\le\dfrac{\pi}{2}\) and the \(x\)-axis, and the reference number is the shortest distance along the unit circle to the \(x\)-axis. Since the radius of the unit circle is by definition 1 unit, the reference number and...
  6. MarkFL

    Challenge Find the ratio of the shaded area to the area of the largest semi-circle

    Consider the following diagram: What fraction of the area of the largest semicircle is the shaded area?
  7. MarkFL

    Practice Logistic Function

    We are given: N(t)=\frac{P}{1+ae^{-bt}} We can see that: \lim_{t\to\infty}N(t)=P And from the graph, we see that \(P=5\), and so we have: N(t)=\frac{5}{1+ae^{-bt}} Because the point \((0,1)\) is on the graph of the function, we know: N(0)=\frac{5}{1+a}=1\implies a=4 And so we now have...
  8. MarkFL

    Practice Half-life of Thorium-232

    They should look like: Each unit on the horizontal axis represents a half-life. A unit on the vertical axis represents \(A_0\).
  9. MarkFL

    Practice Bacteria Population

    The setting have not changed. Testing...
  10. MarkFL

    Practice Find a & b

    Okay, we have: 2=ae^{b} 8=ae^{4b} Hence: a=2e^{-b}=8e^{-4b} e^{-b}=4e^{-4b} e^{3b}=4 b=\frac{2}{3}\ln(2)\implies a=2^{\frac{1}{3}} Thus: y=2^{\frac{1}{3}}e^{\frac{2}{3}\ln(2)x}=2^{\frac{2x+1}{3}}
  11. MarkFL

    Practice Exponential Equations 1

    0=2^x-5 2^x=5 x=\log_2(5)\approx2.321928094887362
  12. MarkFL

    Practice Region Bounded By Two Exponential Functions

    Here is a diagram that should help:
  13. MarkFL

    Practice Values of x

    If we take the natural log of both sides, we get: x<x\ln(10) x-x\ln(10)<0 x(1-\ln(10))<0 x>0 Here is a graph: \(y=10^x\) is in green, and \(y=e^x\) is in blue. Suppose we are given: a^x<b^x where \(a<b\)...can you algebraically solve this inequality?
  14. MarkFL

    Practice Moe and Larry

    One way to solve this would be to graph the two lines and read off the point of intersection: Another way is to equate the two functions and solve for \(t\): 20t+35=10t+60 10t=25 t=\frac{5}{2} We find that this problem is ill-designed in that the value we obtain for \(t\) is not an...
  15. MarkFL

    Practice Functions Have An Inverse?

    Let's look at a graph of the first one: What is it we're looking for to determine if the function has an inverse?
  16. MarkFL

    Practice Exponential Functions l

    Here is a plot of all 3 functions: The one in bold red is \(y=e^x\). The one in green is \(y=e^{-x}\), which as I stated is a reflection of the original across the \(y\)-axis. The one in blue is \(y=-e^x\) which is a reflection of the original across the \(x\)-axis.
  17. MarkFL

    Practice Exponential Function 1

    To follow up: 1.) y=e^{-x} Domain: (-\infty,\infty) Range: (0,\infty) Intercept(s): (0,1) Asymptotes: y=0 2. y=-e^{-x} Domain: (-\infty,\infty) Range: (-\infty,0) Intercept(s): (0,-1) Asymptotes: y=0
  18. MarkFL

    Practice Exponential Function Prove 2

    Consider the following rectangle: This is a "golden rectangle" because: \frac{x}{y}=\varphi where \(\varphi\) is the golden ratio. The golden rectangle has the property such that when a square is cut off (shaded in red) the remaining rectangle is similar to the original. And so we may...
  19. MarkFL

    Practice Exponential Functions

    A. Domain: (-\infty,\infty) Range: (-\infty,3) Intercepts: (0,2),\,(1,0) Aymptote: y=3 B. See what you can do. :)
  20. MarkFL

    Practice Analyze Rational Function

    The only issue I see with your sketch is that the function is negative in between its roots. :) Oh, and the leftmost branch isn't shown.